Triangle Inequality <= Sign Explained

[razored]

http://math.ucsd.edu/~wgarner/math4c/derivations/other/triangleinequal_files/eq0007S.gif

Why did they introduce the <= sign?

I cannot think of any numbers that would violote the =. So why introduce the <?

 

[bucher]

Because if "a" or "b" (not both) is negative, then the answer would be less than the formula with the absolute value of "a" or "b".

 

[razored]

Because if "a" or "b" (not both) is negative, then the answer would be less than the formula with the absolute value of "a" or "b".

That's not true, plugin a=-3 and b=2. In fact, try any set of numbers and you will see.

 

[HallsofIvy]

Okay, I will: |-3+2|^2= 1^2= 1. |-3|+ |2|= 3+2= 5 and 5^2= 25. 1 is definitely less than 25!

Now, what do YOU get? (Or did you do |a+b|² and (a+b)² rather than |a+b|² and (|a|+ |b|)^2?)

 

[Fredrik]

I think (or at least I hope) that the OP is referring to the last line (not the one before). The last one should start with a =.

 

[HallsofIvy]

That is not my interpretation of that particular way of writing mathematics.

If it were |a+b|^2= (a+ b)(a+ b)= a^2+ 2ab+ b^2\le |a|^2+ 2|a||b|+ |b|^2= (|a|+ |b|)^2, in one line, then, yes, the last two are equal. But my understanding of
\begin{array}{cc}|a+ b|^2&amp;= (a+b)(a+b)\\ &amp;= a^2+ 2ab+ b^2\\ &amp;\le |a|^2+ 2|a||b|+ |b|^2 \\ &amp;\le (|a|+ |b|)^2
is that the left side, here |a+ b|^2, is "copied" down the left. That is, it is
\begin{array}{cc}|a+ b|^2&amp;= (a+b)(a+b)\\|a+ b|^2&amp;= a^2+ 2ab+ b^2\\|a+ b|^2&amp;\le |a|^2+ 2|a||b|+ |b|^2 \\|a+ b|^2&amp;\le (|a|+ |b|)^2

 

[Dunkle]

That is not my interpretation of that particular way of writing mathematics.

Your interpretation is correct (but I know you don't need me to tell you that). Most authors write this way, and it even saves ink!

 

[razored]

I think (or at least I hope) that the OP is referring to the last line (not the one before). The last one should start with a =.

The last one should be with <=. I understand why it is there; I was just pissed because I would have never thought to put the <= after putting up absolute values around 2ab.

Thanks for the help.

HallsOfIvy, I do like your interpretation of the math better, more lucid. The GIF I posted is hotlinked from some website I found. Unfortunately, the book I got this problem (spivak) from uses the same notation as the GIF image.